Tham khảo tài liệu 'a course in mathematical statistics phần 8', ngoại ngữ, ngữ pháp tiếng anh phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 370 13 Testing Hypotheses Applications of LR Tests Contingency Tables Goodness-of-Fit Tests Now we turn to a slightly different testing hypotheses problem where the LR is also appropriate. We consider an r. experiment which may result in k possibly different outcomes denoted by Oị j 1 . k. In n independent repetitions of the experiment let pj be the constant probability that each one of the trials will result in the outcome Oj and denote by Xj the number of trials which result in Oj j 1 . k. Then the joint distribution of the X s is the Multinomial distribution that is P X1 x1 . Xk xk n P1X1 pkk v x1 xk where xj 0 j 1 . k 1xj n and Q 0 P1 . Pk Pj 0 j 1 . k XPj 1 j 1 We may suspect that the p s have certain specified values for example in the case of a die the die may be balanced. We then formulate this as a hypothesis and proceed to test it on the basis of the data. More generally we may want to test the hypothesis that 9 lies in a subset rn of Q. Consider the case that H 0 e ffl 0O p10 . pk0 . Then under ffl n x x _ c n x Ị p10 pk0 while under Q l q n__px1 p xk -MQ p1 pk v x1 xk where pj xj n are the MLE s of pj j 1 . k see Example 11 Chapter 12 . Therefore t A n n p0 7 T V xj 7 and H is rejected if -2logA C. The constant C is determined by the fact that -2log A is asymptotically X2k-1 distributed under H as it can be shown on the basis of Theorem 6 and the desired level of significance a. Now consider r events A i 1 . r which form a partition of the sample space S and let B j 1 . s be another partition of S. Let p7 P Ai n Bj and let p . . Ep7 . Applications of LR Tests Contingency Tables Goodness-of-Fit Tests 371 Then clearly P. P Ai Pj P Bj and r s r s t P t tt PiJ 1. i 1 J 1 i 1 J 1 Furthermore the events A1 . Ar and B1 . Bs are independent if and only if Pij i 1 . r J 1 . s. A situation where this set-up is appropriate is the following Certain experimental units are classified according to two characteristics denoted by A
